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Structure Theory of Semisimple Lie Groups

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Structure Theory of Semisimple Lie Groups Proceedings of Symposia in Pure Mathematics Volume 61 (1997), pp. 1–27 Structure Theory of Semisimple Lie Groups A. W. Knapp This article provides a review of the elementary theory of semisimple Lie al- gebras and Lie groups. It is essentially a summary of much of [K3]. The four sections treat complex semisimple Lie algebras, finite-dimensional representations of complex semisimple Lie algebras, compact Lie groups and real forms of complex Lie algebras, and structure theory of noncompact semisimple groups. 1. Complex Semisimple Lie Algebras This section deals with the structure theory of complex semisimple Lie algebras. Some references for this material are [He], [Hu], [J], [K1], [K3], and [V]. Let g be a finite-dimensional Lie algebra. For the moment we shall allow the underlying field to be R or C, but shortly we shall restrict to Lie algebras over C. Semisimple Lie algebras are defined as follows. Let rad g be the sum of all the solvable ideals in g. The sum of two solvable ideals is a solvable ideal [K3, §I.2], and the finite-dimensionality of g makes rad g a solvable ideal. We say that g is semisimple if rad g =0. Within g, let ad X be the linear transformation given by (ad X)Z =[X, Z]. The Killing form is the symmetric bilinear form on g defined by B(X, Y )= Tr(ad X ad Y ). It is invariant in the sense that B([X, Y ],Z)=B(X, [Y,Z]) for all X, Y, Z in g. Theorem 1.1 (Cartan’s criterion for semisimplicity). The Lie algebra g is semisimple if and only if B is nondegenerate. Reference. [K3, Theorem 1.42]. The Lie algebra g is said to be simple if g is nonabelian and g has no proper nonzero ideals. In this case, [g, g]=g. Semisimple Lie algebras and simple Lie algebras are related as in the following theorem. Theorem 1.2. The Lie algebra g is semisimple if and only if g is the direct sum of simple ideals. In this case there are no other simple ideals, the direct sum decomposition is unique up to the order of the summands, and every ideal is the sum of some subset of the simple ideals. Also in this case, [g, g]=g. 1991 Mathematics Subject Classification. Primary 17B20, 20G05, 22E15. c 1997 A. W. Knapp 1 2 A. W. KNAPP Reference. [K3, Theorem 1.51]. For the remainder of this section, g will always denote a semisimple Lie algebra, and the underlying field will be C. The dual of a vector space V will be denoted V ∗. We discuss root-space decompositions. For our semisimple Lie algebra g, these are decompositions of the form g = h ⊕ gα. α∈∆ Here h is a Cartan subalgebra, defined in any of three equivalent ways [K3, §§II.2–3] as (a) (usual definition) a nilpotent subalgebra h whose normalizer satisfies Ng(h)=h, (b) (constructive definition) the generalized eigenspace for 0 eigenvalue for ad X with X regular (i.e., characteristic polynomial det(λ1 − ad X) is such that the lowest-order nonzero coefficient is nonzero on X), (c) (special definition for g semisimple) a maximal abelian subspace of g in which every ad H, H ∈ h, is diagonable. ∗ The elements α ∈ h are roots, and the gα’s are root spaces, the α’s being defined as the nonzero elements of h∗ such that gα = {X ∈ g | [H, X]=α(H)X for all H ∈ h} is nonzero. Let ∆ be the set of all roots. This is a finite set. We recall the the classical examples of root-space decompositions [K3, §II.1]. Example 1. g = sl(n, C)={n-by-n complex matrices of trace 0}. The Cartan subalgebra is h = {diagonal matrices in g}. Let 1in(i, j)th place Eij = 0 elsewhere. ∗ Let ei ∈ h be defined by h1 . ei .. = hi. hn Then each H ∈ h satisfies (ad H)Eij =[H, Eij]=(ei(H) − ej(H))Eij. So Eij is a simultaneous eigenvector for all ad H, with eigenvalue ei(H) − ej(H). We conclude that (a) h is a Cartan subalgebra, (b) the roots are the (ei − ej)’s for i = j, C (c) gei−ej = Eij. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 3 Example 2. g = so(2n +1, C)={n-by-n skew-symmetric complex matrices}. For this example one proceeds similarly. Let h = {H ∈ so(2n +1, C) | H = matrix below}. Here H is block diagonal with n 2-by-2 blocks and one 1-by-1 block, 0 ih the jth 2-by-2 block is j , −ihj 0 the 1-by-1 block is just (0). Let ej(above matrix H)=hj for 1 ≤ j ≤ n. Then ∆={±ei ± ej with i = j}∪{±ek}. Formulas for the root vectors Eα may be found in [K3, §II.1]. Example 3. g = sp(n, C). This is the Lie algebra of all 2n-by-2n complex matrices X such that 0 I XtJ + JX =0, where J = . −I 0 For this example the Cartan subalgebra h is the set of all matrices H of the form h1 . .. hn H = − h1 . .. −hn Let ej(above matrix H)=hj for 1 ≤ j ≤ n. Then ∆={±ei ± ej with i = j}∪{±2ek}. Formulas for the root vectors Eα may again be found in [K3, §II.1]. Example 4. g = so(2n, C). This example is similar to so(2n +1, C) but without the (2n +1)st entry. The set of roots is ∆={±ei ± ej with i = j}. We return to the discussion of general semisimple Lie algebras g. The following are some elementary properties of root-space decompositions: (a) [gα, gβ] ⊆ gα+β. (b) If α and β are in ∆ ∪{0} and α + β = 0, then B(gα, gβ)=0. (c) If α is in ∆, then B is nonsingular on gα × g−α. (d) If α is in ∆, then so is −α. (e) B|h×h is nondegenerate. Define Hα to be the element of h paired with α. (f) ∆ spans h∗. See [K3, §II.4]. We isolate some deeper properties of root-space decompositions as a theorem. 4 A. W. KNAPP Theorem 1.3. Root-space decompositions have the following properties: (a) If α is in ∆, then dim gα =1. (b) If α is in ∆, then nα is not in ∆ for any integer n ≥ 2. (c) [gα, gβ]=gα+β if α + β =0 . (d) The real subspace h0 of h on which all roots are real is a real form of h, and | | ∗ B h0×h0 is an inner product. Transfer B h0×h0 to the real span h0 of the roots, obtaining ·, · and |·|2. Reference. [K3, §II.4]. Let us now consider root strings. By definition the α string containing β (for α ∈ ∆, β ∈ ∆ ∪{0}) consists of all members of ∆ ∪{0} of the form β + nα with 2β,α n ∈ Z. The n’s in question form an interval with −p ≤ n ≤ q and p − q = . |α|2 Here p − q is a measure of how centered β is in the root string. When p − q is 0, β is exactly in the center. When p − q is large and positive, β is close to the end 2β,α β + qα of the root string. In any event, it follows that is always an integer. |α|2 A consequence of the form of root strings is that if α is in ∆, then the orthogonal ∗ transformation of h0 given by 2ϕ, α s (ϕ)=ϕ − α α |α|2 carries ∆ into itself. The linear transformation sα is called the root reflection in α. An abstract root system is a finite set ∆ of nonzero elements in a real inner product space V such that (a) ∆ spans V , (b) all sα for α ∈ ∆ carry ∆ to itself, 2β,α (c) is an integer whenever α and β are in ∆. |α|2 We say that an abstract root system is reduced if α ∈ ∆ implies 2α/∈ ∆. The relevance of these notions to semisimple Lie algebras is that the root system of a complex semisimple Lie algebra g with respect to a Cartan subalgebra h forms ∗ a reduced abstract root system in h0. See [K3, Theorem 2.42]. There are four kinds of classical reduced root systems: ⊥ n+1 Rn+1 { − | } An has V = i=1 ei in and ∆ = ei ej i = j . The system An arises from sl(n +1, C). n Bn has V = R and ∆ = {±ei ± ej | i = j}∪{±ek}. The system Bn arises from so(2n +1, C). n Cn has V = R and ∆ = {±ei ± ej | i = j}∪{±2ek}. The system Cn arises from sp(n, C). n Dn has V = R and ∆ = {±ei ± ej | i = j}. The system Dn arises from so(2n, C). We say that an abstract root system ∆ is reducible if∆=∆ ∪ ∆ with ∆ ⊥ ∆. Otherwise ∆ is irreducible. Theorem 1.4. A semisimple Lie algebra g is simple if and only if the corre- sponding root system ∆ is irreducible. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 5 Reference. [K3, Proposition 2.44]. Now we introduce the notions of lexicographic ordering and positive roots for an abstract root system. The construction is as follows. Let ϕ1,...,ϕm be a spanning set for V . Define ϕ to be positive (written ϕ>0) if there exists an index k such that ϕ, ϕi = 0 for 1 ≤ i ≤ k − 1 and ϕ, ϕk > 0. The corresponding lexicographic ordering has ϕ>ψif ϕ − ψ is positive. Fix such an ordering. Call the root α simple if α>0 and if α does not decompose as α = β1 + β2 with β1 and β2 both positive roots.
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